Mixed-precision amortized adjoint

What it measures: the gradient accuracy and backward cost of mixed_precision_block_thomas(implicit_adjoint=True) — float32 factors reused (transposed) for the adjoint refinement — against the default unrolled differentiation of the refinement loop and against the bare low-precision gradient. See the Mixed-precision iterative refinement guide.

Reproduce:

python benchmarks/benchmark_mixed_precision_adjoint.py --json

Record: benchmarks/results/mixed_precision_adjoint.json (float32 factors, float64 working precision, two refinement sweeps, CPU).

Gradient accuracy across conditioning

Relative error against the gradient of the exact float64 solve. The implicit adjoint sits at working precision across the entire dominance sweep — the gradient inherits the refined forward error, not the factorization precision [CH18] — while the unrefined gradient carries float32-level error:

dominance	implicit (2 sweeps)	bare fp32 (0 sweeps)	unrolled (2 sweeps)
6.0	3.56e-16	1.73e-7	3.01e-16
4.0	3.25e-16	1.19e-7	2.70e-16
2.0	2.93e-16	1.06e-7	3.21e-16
1.5	4.59e-16	1.13e-7	3.57e-16
1.2	4.52e-16	9.13e-8	4.80e-16

Backward cost

Unrolled differentiation is equally accurate when the refinement converges — the difference is cost. The custom VJP’s backward is refinement sweeps on the transposed factors, not a taped loop through a differentiated factorization:

N, m	unrolled temp	implicit temp	unrolled warm	implicit warm
64, 8	6.58 MiB	0.06 MiB	15.3 ms	4.2 ms
256, 8	104.3 MiB	0.22 MiB	257.8 ms	37.8 ms
256, 16	401.1 MiB	0.59 MiB	1274.5 ms	31.9 ms

At N=256, m=16 the implicit adjoint uses 680× less reverse-mode scratch and runs 40× faster, with roughly half the compile time.

Platform contrast

The accuracy dichotomy is platform-independent: on the A4000 the implicit gradient sits at 3.1e-16 with bare fp32 at 2.0e-7, identical to CPU (benchmarks/results/gpu/mixed_precision_adjoint.json). The backward cost advantage is platform-dependent: 238×/5.9× (memory/time) on CPU at N=128, m=8, but 2×/1.3× on the GPU, whose executor absorbs the tape and factor-VJP far better at this size (and compiles 32 s vs 302 s). Cost claims are therefore stated per platform; the accuracy theorem is not.